|
This reminds rather heavily of a section in Theory of Poker where Sklansky explains that we should randomise our Hero Q betting and Villain K calling with the exact percentage one that is based on the size of the bet.
Specifically you should make the odds against you bluffing the same as your pot odds. So if the pot is $200 and the bet size is $100 your pot odds are 3-to-1 and the odds against you bluffing should be 3-to-1 against. So you should be bluffing 25% of the time.
In the AKQ example when you are Hero with Q you should then bluff 33% of the time. That's because, when you bet, 25% of your bets should be bluffs - and the 75% of them should not be bluffs. They should be A betting for value. So 75% of all bets are 100% of all A - and the corresponding percentage of Q is 25/75=33.33%.
I've done up a little spreadsheet with all this preprogrammed in and I notice a few oddities (after weeding out too many small errors).
When the Q Bet % is the ideal one defined by game theory the K Call % can be anything and the EV for both sides remains the same. Exact same.
In this game, apparently, if Hero is playing correctly Villain can make no difference to the outcome by picking a calling frequency - and if Hero is not playing correctly Villain's best strategy is exploitative (either 100% call or 0% call).
It's only when he thinks hero might be close to right that he can minimise his losses by picking the right calling frequency.
In ToP Sklansky seems to have done lots of EV calculations for the bluffing bit, but not so many for calling the bluffing. He says that when bluffing you should match pot odds with odds against bluffing and when calling bluff he says you should do the same - but he does it differently. For bluffing he considers the pot including the call. For calling the bluff he considers the pot without a call. His experimental EV calculations leads him aright with the bluffing part.
Easy way to put that in a spreadsheet is: bet/(pot+bet)
The result is not how often your bet is a bluff, but how often you should bluff with your Q.
In the section on calling bluffs he's a bit confused and goes on about finding it in a similar manner and suggests that a $100 pot and $20 bet means you should call 5 times and fold once. It's not really that similar when the similar situation is one in which he would call 6 times and fold once. It's also incorrect as my spreadsheet EV has shown me. I kept wondering why by reducing the Hero Q bluffing % when my Villain K calling % should provoke an equilibrium was higher EV than the "optimal". I guessed that my formula was wrong and experimented until I found one that works.
I do not yet know why mathematically it is so, but I can note experimentally that the unexploitable strategy for K is to call (pot-bet)/(pot+bet) of the time when facing a bet.
Anyway, the real reason I was playing around with spreadsheets was because I wanted to find if there was a specific bet size that maximised EV for Hero. And there is. It's around 41.43% of the pot, which you then bluff with 29.29% of the time.
If you want to make it a party game, suggest fixing the bet size at 50% of the pot, because there both the unexploitable Hero Q bluff % and the unexploitable Villain K call % is 33.33% (yeah, exactly one third). Given a wrist watch (can be split into thirds as easily as halves or quarters) or some other object that gives true random thirds you would have a certain EV edge - as long as you make sure you play at least as many times as Hero as you play as Villain. Because villain really isn't in for much EV.
Another possible fixed bet size is 1/3 pot where Q bluff % is 25% and K call % is 50%.
1/4 pot: Q bluff % 20 - K call % 60
2/3 pot: Q bluff % 40 - K call % 20
full pot: Q bluff % 50 - K call % 0
With full pot it's 0 EV both ways. At 41.43% the EV is +2.86% of the pot for Hero per game. At 1/2 pot and 1/3 pot it is 2.78%, at 1/4 pot it is 2.50%, at 2/3 pot it is 2.22%.
Problem with playing full pot (and not caring which side you're on) is that optimal strategy is to always fold the K. That way it's very easy for someone to spot what you're doing and simply copy your actions. While you will still have an edge when you play Villain and someone tries to bluff (or not), if someone makes you play Hero you get the stress of doing the randomisation right while your opponent can easily play perfect against you.
|